====== Feb D-Dudes Meeting ======

Some brief "cliff notes" for my Feb talk to D-Dudes:

====== Last time ======

Had assumptions:
  - max pairwise interactions
  - one type of individual
  - <del>only focal individual affected</del>((Assumption has since been removed.))
  - <del>partner type ignored</del>((Assumption has since been removed.))

====== Since then ======

  * removed two assumptions
  * even with just two remaining assumptions can **only** get logistic equation (demo)
  * what is "lowest-order" correction?
  * keep pairwise assumption, but let there be two types
  * approached from two directions: bottom-up and top-down
  * brought them as close together as possible.  Now need to test.  Anything else I should do before diving in?

====== Bottom-up approach ======

New assumptions:
  - max pairwise interactions
  - two types of individuals (in one species/population)
  - separation of timescales: physiology much faster than ecology

Separation of timescales necessary trick to reduce system of two equations back down to single equation.  Need ecological rate constants (birth/death rates) to be different for two states.  Seems reasonable if, eg. physiological states are healthy/sick.

Eventually arrive at equation \(n' = n g(n)\) where \(g(n)\) is complicated and depends on rate constants.

====== Top-down approach ======

Ideally, want "higher-order correction" to look similar to logistic.  Good choice seems to be \(n'=r n a(n) (1-n/K)\) where \(a(n)=1-A + A n / (n+n_c) = (n-n_A)/(n+n_c)\).  \(a(n)\) can be interpreted as Allee effect (depression of growth rate for small \(n\)).

If possible, want \(r\) to be constant, representing //intrinsic growth rate// excepting Allee effect.  So take \(r=g(n)/[a(n)(1-n/K)]\) and make \(r\) as "flat" as possible, at least for small \(n\).  Can be done by choosing parameters \((A, n_c, K)\) so that low order terms in Taylor expansion of \(r\) are null.

====== What's next? ======

  * tests
    * generate random rate constants
    * check if approximation should fit
      - Is there a good timescale separation?
      - Is population in regime where \(r\) nearly constant?
    * check if it does
      - run full model
      - run approximate model
      - compare (Q: How to measure fit?)
    * repeat

Best case: it fits even better than expected.  Can then claim that logistic with Allee factor is natural generalization of logistic.

Ok case: it mostly fits when it should and doesn't when it shouldn't.  Can then claim that logistic+Allee is good generalization under specific conditions.

Worst case: it doesn't fit.  Pack it in.  What could have gone wrong?  Maybe bad choice of \(a(n)\)?

====== Follow-up ======

Michael made a good point: it would be great to find a really counter-intuitive result to make people sit up and take notice.  His idea is to find a set of parameters that yields a strong Allee effect (\(A>1\)) and then compare it against an "equivalent" logistic model.  The logistic model is inherently stable at \(n=K\) so this would show how going to a higher resolution (two-types) reveals a critical population threshold that would otherwise be missed.  The coarse logistic model can make serious/dangerous mistakes.

Michael also suggested another counter-intuitive case to look for: Check if it can ever be the case that the approximate birthrate $r$ can ever be smaller than both the net birthrates ($\alpha-\delta$) of the two sub-types.  That would be weird.  How to check?

